4.4 向量自回归模型（Vector Autoregression）

以上三个章节我们讨论的都是单变量问题，现实世界往往更为复杂，这时我们将把AR模型从单变量拓展到多变量，多变量模型的特点是存在几组并行的时间序列，并且这些时间之间互相影响。

本章讨论的模型就叫做向量自回归模型（VAR)，考虑一个三组并行时间序列数据的场景，在二阶的条件下，公式如下：

y_{1,t} = \phi_{01} + \phi_{11,1} * y_{1,t -1} + \phi_{12,1} * y_{2,t -1} + \phi_{13,1} * y_{3,t -1} + \phi_{11,2} * y_{1,t -2} + \phi_{12,2} * y_{2,t -2} + \phi_{13,2} * y_{3,t -2} \\ y_{2,t} = \phi_{02} + \phi_{21,1} * y_{1,t -1} + \phi_{22,1} * y_{2,t -1} + \phi_{23,1} * y_{3,t -1} + \phi_{21,2} * y_{1,t -2} + \phi_{22,2} * y_{2,t -2} + \phi_{23,2} * y_{3,t -2} \\ y_{3,t} = \phi_{03} + \phi_{31,1} * y_{1,t -1} + \phi_{32,1} * y_{2,t -1} + \phi_{33,1} * y_{3,t -1} + \phi_{31,2} * y_{1,t -2} + \phi_{32,2} * y_{2,t -2} + \phi_{33,2} * y_{3,t -2}

可以看到在AR模型的基础上，每一组时间序列的y值都加入了其他两组时间序列值作为模型因子。熟悉线性代数的读者可能已经发现，上面的三个公式可以写成向量乘法的形式，这也是VAR模型名字的由来，写作向量乘法的公式和前面学习的AR模型时完全一致的，在这个公式中，y和 $\phi_0$ 是 $3\times1$ 的向量， $\phi_1, \phi_2$ 是 $3\times3$ 的矩阵。

y = \phi_0 + \phi_1 × y_{t -1} + \phi_2 × y_{t -2}

从公式中也能看到随着阶数上升，VAR模型的变量增加很快，因此在使用时只有当期待存在不同时间序列互相影响的关系时才尝试这种方法。VAR在某些场景下十分有用，

测试某个变量是否影响其他变量
大量变量需要被预测，而分析师没有太多领域知识
决定某个预测值在多大程度上是由潜在的因果性导致的

python实战部分

import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
from statsmodels.tsa.stattools import adfuller
from statsmodels.tsa.api import VAR
from statsmodels.tsa.stattools import grangercausalitytests
from statsmodels.tsa.vector_ar.vecm import coint_johansen
from statsmodels.stats.stattools import durbin_watson
%matplotlib inline

# step1：导入数据，这是一个有八个指标的经济数据
df = pd.read_csv('data/Raotbl6.csv', parse_dates=['date'], index_col='date')
df.head()

# 画出八个时间序列的图像
fig, axes = plt.subplots(nrows=4, ncols=2,figsize=(10,6))
for i, ax in enumerate(axes.flatten()):
    data = df[df.columns[i]]
    ax.plot(data, color='red', linewidth=1)
    ax.set_title(df.columns[i])
plt.tight_layout();

# step2：Granger’s Causality Test ， 检验不同序列之间存在互相影响
maxlag=12
test='ssr_chi2test'
variables=df.columns
def grangers_causation_matrix(data, variables, test='ssr_chi2test', verbose=False):    
    df = pd.DataFrame(np.zeros((len(variables), len(variables))), columns=variables, index=variables)
    for c in df.columns:
        for r in df.index:
            test_result = grangercausalitytests(data[[r, c]], maxlag=maxlag, verbose=False)
            p_values = [round(test_result[i+1][0][test][1],4) for i in range(maxlag)]
            min_p_value = np.min(p_values)
            df.loc[r, c] = min_p_value
    df.columns = [var + '_x' for var in variables]
    df.index = [var + '_y' for var in variables]
    return df

grangers_causation_matrix(df, variables = df.columns)

在输出结果中，index以y结尾，表示响应变量，column以x结尾，表示预测变量，如果p值小于0.05表明存在格兰杰因果性。因此根据检验数据，完全有理由使用VAR模型。

# step3：ADF测试，检验单个变量是否平稳
def adfuller_test(series, signif=0.05, name='', verbose=False):
    r = adfuller(series, autolag='AIC')
    output = {'test_statistic':round(r[0], 4), 'pvalue':round(r[1], 4), 'n_lags':round(r[2], 4), 'n_obs':r[3]}
    p_value = output['pvalue'] 
    def adjust(val, length= 6): return str(val).ljust(length)

    # Print Summary
    print(f'    Augmented Dickey-Fuller Test on "{name}"', "\n   ", '-'*47)
    print(f' Null Hypothesis: Data has unit root. Non-Stationary.')
    print(f' Significance Level    = {signif}')
    print(f' Test Statistic        = {output["test_statistic"]}')
    print(f' No. Lags Chosen       = {output["n_lags"]}')

    for key,val in r[4].items():
        print(f' Critical value {adjust(key)} = {round(val, 3)}')

    if p_value <= signif:
        print(f" => P-Value = {p_value}. Rejecting Null Hypothesis.")
        print(f" => Series is Stationary.")
    else:
        print(f" => P-Value = {p_value}. Weak evidence to reject the Null Hypothesis.")
        print(f" => Series is Non-Stationary.")

for name, column in df.iteritems():
    adfuller_test(column, name=column.name)
    print('\n')

    Augmented Dickey-Fuller Test on "rgnp" 
    -----------------------------------------------
 Null Hypothesis: Data has unit root. Non-Stationary.
 Significance Level    = 0.05
 Test Statistic        = 0.6419
 No. Lags Chosen       = 2
 Critical value 1%     = -3.486
 Critical value 5%     = -2.886
 Critical value 10%    = -2.58
 => P-Value = 0.9886. Weak evidence to reject the Null Hypothesis.
 => Series is Non-Stationary.

Augmented Dickey-Fuller Test on "pgnp" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = 1.2743 No. Lags Chosen = 1 Critical value 1% = -3.486 Critical value 5% = -2.886 Critical value 10% = -2.58 => P-Value = 0.9965. Weak evidence to reject the Null Hypothesis. => Series is Non-Stationary.

Augmented Dickey-Fuller Test on "ulc" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = 1.3967 No. Lags Chosen = 2 Critical value 1% = -3.486 Critical value 5% = -2.886 Critical value 10% = -2.58 => P-Value = 0.9971. Weak evidence to reject the Null Hypothesis. => Series is Non-Stationary.

Augmented Dickey-Fuller Test on "gdfco" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = 0.5762 No. Lags Chosen = 5 Critical value 1% = -3.488 Critical value 5% = -2.887 Critical value 10% = -2.58 => P-Value = 0.987. Weak evidence to reject the Null Hypothesis. => Series is Non-Stationary.

Augmented Dickey-Fuller Test on "gdf" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = 1.1129 No. Lags Chosen = 7 Critical value 1% = -3.489 Critical value 5% = -2.887 Critical value 10% = -2.58 => P-Value = 0.9953. Weak evidence to reject the Null Hypothesis. => Series is Non-Stationary.

Augmented Dickey-Fuller Test on "gdfim" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -0.1987 No. Lags Chosen = 1 Critical value 1% = -3.486 Critical value 5% = -2.886 Critical value 10% = -2.58 => P-Value = 0.9387. Weak evidence to reject the Null Hypothesis. => Series is Non-Stationary.

Augmented Dickey-Fuller Test on "gdfcf" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = 1.6693 No. Lags Chosen = 9 Critical value 1% = -3.49 Critical value 5% = -2.887 Critical value 10% = -2.581 => P-Value = 0.9981. Weak evidence to reject the Null Hypothesis. => Series is Non-Stationary.

Augmented Dickey-Fuller Test on "gdfce" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -0.8159 No. Lags Chosen = 13 Critical value 1% = -3.492 Critical value 5% = -2.888 Critical value 10% = -2.581 => P-Value = 0.8144. Weak evidence to reject the Null Hypothesis. => Series is Non-Stationary.

没有一个变量具有平稳性，提示我们需要进一步进行协整检验（cointegration test）。一般来说进行协整检验的步骤如下： 1.单独检验单个变量是否平稳，使用ADF test, KPSS test, PP test等方法。 2.如果发现单个序列不平稳，则需要进一步进行协整检验。进行协整检验的目的是当每个变量本身不平稳，有可能他们在某些线性组合下是平稳的。如果两个时间序列是协整的，则表明他们具有长期的，统计学显著的关联，使用Johansen, Engle-Granger, and Phillips-Ouliaris等方法。

# step4: 协整检验，检验多变量平稳性
def cointegration_test(df, alpha=0.05): 
    out = coint_johansen(df,-1,5)
    d = {'0.90':0, '0.95':1, '0.99':2}
    traces = out.lr1
    cvts = out.cvt[:, d[str(1-alpha)]]
    def adjust(val, length= 6): return str(val).ljust(length)

    # Summary
    print('Name   ::  Test Stat > C(95%)    =>   Signif  \n', '--'*20)
    for col, trace, cvt in zip(df.columns, traces, cvts):
        print(adjust(col), ':: ', adjust(round(trace,2), 9), ">", adjust(cvt, 8), ' =>  ' , trace > cvt)

cointegration_test(df)

Name   ::  Test Stat > C(95%)    =>   Signif  
 ----------------------------------------
rgnp   ::  248.0     > 143.6691  =>   True
pgnp   ::  183.12    > 111.7797  =>   True
ulc    ::  130.01    > 83.9383   =>   True
gdfco  ::  85.28     > 60.0627   =>   True
gdf    ::  55.05     > 40.1749   =>   True
gdfim  ::  31.59     > 24.2761   =>   True
gdfcf  ::  14.06     > 12.3212   =>   True
gdfce  ::  0.45      > 4.1296    =>   False

# step5：划分训练集和测试集
nobs = 4  # 最后四个时间点作为测试集
df_train, df_test = df[0:-nobs], df[-nobs:]

# step6：使用VAR之间，先差分处理使单个变量变得平稳
df_differenced = df_train.diff().dropna()
for name, column in df_differenced.iteritems():
    adfuller_test(column, name=column.name)
    print('\n')

# 一阶差分后仍然有些变量不平稳，进行二次差分
df_differenced = df_differenced.diff().dropna()

for name, column in df_differenced.iteritems():
    adfuller_test(column, name=column.name)
    print('\n')

    Augmented Dickey-Fuller Test on "rgnp" 
    -----------------------------------------------
 Null Hypothesis: Data has unit root. Non-Stationary.
 Significance Level    = 0.05
 Test Statistic        = -9.0123
 No. Lags Chosen       = 2
 Critical value 1%     = -3.489
 Critical value 5%     = -2.887
 Critical value 10%    = -2.58
 => P-Value = 0.0. Rejecting Null Hypothesis.
 => Series is Stationary.

Augmented Dickey-Fuller Test on "pgnp" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -10.9813 No. Lags Chosen = 0 Critical value 1% = -3.488 Critical value 5% = -2.887 Critical value 10% = -2.58 => P-Value = 0.0. Rejecting Null Hypothesis. => Series is Stationary.

Augmented Dickey-Fuller Test on "ulc" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -8.769 No. Lags Chosen = 2 Critical value 1% = -3.489 Critical value 5% = -2.887 Critical value 10% = -2.58 => P-Value = 0.0. Rejecting Null Hypothesis. => Series is Stationary.

Augmented Dickey-Fuller Test on "gdfco" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -7.9102 No. Lags Chosen = 3 Critical value 1% = -3.49 Critical value 5% = -2.887 Critical value 10% = -2.581 => P-Value = 0.0. Rejecting Null Hypothesis. => Series is Stationary.

Augmented Dickey-Fuller Test on "gdf" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -10.0351 No. Lags Chosen = 1 Critical value 1% = -3.489 Critical value 5% = -2.887 Critical value 10% = -2.58 => P-Value = 0.0. Rejecting Null Hypothesis. => Series is Stationary.

Augmented Dickey-Fuller Test on "gdfim" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -9.4059 No. Lags Chosen = 1 Critical value 1% = -3.489 Critical value 5% = -2.887 Critical value 10% = -2.58 => P-Value = 0.0. Rejecting Null Hypothesis. => Series is Stationary.

Augmented Dickey-Fuller Test on "gdfcf" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -6.922 No. Lags Chosen = 5 Critical value 1% = -3.491 Critical value 5% = -2.888 Critical value 10% = -2.581 => P-Value = 0.0. Rejecting Null Hypothesis. => Series is Stationary.

Augmented Dickey-Fuller Test on "gdfce" ----------------------------------------------- Null Hypothesis: Data has unit root. Non-Stationary. Significance Level = 0.05 Test Statistic = -5.1732 No. Lags Chosen = 8 Critical value 1% = -3.492 Critical value 5% = -2.889 Critical value 10% = -2.581 => P-Value = 0.0. Rejecting Null Hypothesis. => Series is Stationary.

# step7：选择模型阶数并训练，根据AIC值，lag=4时达到局部最优
model = VAR(df_differenced)
for i in [1,2,3,4,5,6,7,8,9]:
    result = model.fit(i)
    print('Lag Order =', i)
    print('AIC : ', result.aic, '\n')

Lag Order = 1
AIC :  -1.3679402315450668 

Lag Order = 2
AIC :  -1.621237394447824 

Lag Order = 3
AIC :  -1.76580083870128 

Lag Order = 4
AIC :  -2.000735164470319 

Lag Order = 5
AIC :  -1.9619535608363936 

Lag Order = 6
AIC :  -2.3303386524829053 

Lag Order = 7
AIC :  -2.592331352347122 

Lag Order = 8
AIC :  -3.3172619764582087 

Lag Order = 9
AIC :  -4.804763125958635

# 选择lag=4拟合模型
model_fitted = model.fit(4)
model_fitted.summary()

# step8：durbin watson test，检验残差项中是否还存在相关性，这一步的目的是确保模型已经解释了数据中所有的方差和模式
out = durbin_watson(model_fitted.resid)
for col, val in zip(df.columns, out):
    print(adjust(col), ':', round(val, 2))  # 检验值越接近2，说明模型越好

rgnp   : 2.09
pgnp   : 2.02
ulc    : 2.17
gdfco  : 2.05
gdf    : 2.25
gdfim  : 1.99
gdfcf  : 2.2
gdfce  : 2.17

# step9：模型已经足够使用了，下一步进行预测
lag_order = model_fitted.k_ar
forecast_input = df_differenced.values[-lag_order:]
fc = model_fitted.forecast(y=forecast_input, steps=nobs)
df_forecast = pd.DataFrame(fc, index=df.index[-nobs:], columns=df.columns + '_2d')
df_forecast

# step10：将差分后的值还原为原数据
def invert_transformation(df_train, df_forecast):
    df_fc = df_forecast.copy()
    columns = df_train.columns
    for col in columns: 
        # 写一个长度为6的数列，用纸笔写出差分的计算过程，可以帮助理解下面这两行还原过程
        df_fc[str(col)+'_1d'] = (df_train[col].iloc[-1]-df_train[col].iloc[-2]) + df_fc[str(col)+'_2d'].cumsum()
        df_fc[str(col)+'_forecast'] = df_train[col].iloc[-1] + df_fc[str(col)+'_1d'].cumsum()
    return df_fc

df_results = invert_transformation(df_train, df_forecast)  
df_results.loc[:, ['rgnp_forecast', 'pgnp_forecast', 'ulc_forecast', 'gdfco_forecast',
                   'gdf_forecast', 'gdfim_forecast', 'gdfcf_forecast', 'gdfce_forecast']]

fig, axes = plt.subplots(nrows=int(len(df.columns)/2), ncols=2, dpi=150, figsize=(10,10))
for i, (col,ax) in enumerate(zip(df.columns, axes.flatten())):
    df_results[col+'_forecast'].plot(legend=True, ax=ax).autoscale(axis='x',tight=True)
    df_test[col][-nobs:].plot(legend=True, ax=ax);
    ax.set_title(col + ": Forecast vs Actuals")
    ax.xaxis.set_ticks_position('none')
    ax.yaxis.set_ticks_position('none')
    ax.spines["top"].set_alpha(0)
    ax.tick_params(labelsize=6)

plt.tight_layout();

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